Question

f(x)=(lnx)^x
f'(e)=?

Answer 1

$y=(lnx)^{x}\\\\lny=ln(lnx)^{x}\\\\(lny)'=(x\cdot ln(lnx))'\\\\\frac{y'}{y}=ln(lnx)+x\cdot \frac{1}{lnx}\cdot \frac{1}{x}\\\\y'=y\cdot (ln(lnx)+\frac{1}{lnx})\; ;\; \; \; \; y=(lnx)^{x}\\\\y'=(lnx)^{x}\cdot (ln(lnx)+\frac{1}{lnx})$

$y'(e)=(lne)^{e}(ln(lne)+\frac{1}{lne})=1*(ln1+\frac{1}{1})=1*(0+1)=1$